TRAVELLING WAVE SOLUTIONS IN NONLOCAL REACTION–DIFFUSION SYSTEMS WITH DELAYS AND APPLICATIONS

Abstract This paper deals with two-species convolution diffusion-competition models of Lotka–Volterra type with delays which describe more accurate information than the Laplacian diffusion-competition models. We first investigate the existence of travelling wave solutions of a class of nonlocal convolution diffusion systems with weak quasimonotonicity or weak exponential quasimonotonicity by a cross-iteration technique and Schauder’s fixed point theorem. When the results are applied to the convolution diffusion-competition models with delays, we establish the existence of travelling wave solutions as well as asymptotic behaviour.

[1]  Jack Carr,et al.  Uniqueness of travelling waves for nonlocal monostable equations , 2004 .

[2]  Wan-Tong Li,et al.  Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays , 2006 .

[3]  Van Vuuren,et al.  The existence of travelling plane waves in a general class of competition-diffusion systems , 1995 .

[4]  Xingfu Zou,et al.  Traveling Wave Fronts of Reaction-Diffusion Systems with Delay , 2001 .

[5]  Shigui Ruan,et al.  Persistence and Extinction in Two Species Reaction–Diffusion Systems with Delays , 1999 .

[6]  Wan-Tong Li,et al.  Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems , 2006 .

[7]  Shuxia Pan,et al.  Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity , 2008 .

[8]  Wan-Tong Li,et al.  Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications , 2009 .

[9]  J Umbanhowar,et al.  Non-local concepts and models in biology. , 2001, Journal of theoretical biology.

[10]  Klaus W. Schaaf.  Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations , 1987 .

[11]  P. Bates,et al.  Traveling Waves in a Convolution Model for Phase Transitions , 1997 .

[12]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .

[13]  Jerome Coville,et al.  Nonlocal anisotropic dispersal with monostable nonlinearity , 2008, 1106.4531.

[14]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[15]  Wenzhang Huang,et al.  Travelling waves for delayed reaction–diffusion equations with global response , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  J. Medlock,et al.  Spreading disease: integro-differential equations old and new. , 2003, Mathematical biosciences.

[17]  A. Ōkubo,et al.  On the spatial spread of the grey squirrel in Britain , 1989, Proceedings of the Royal Society of London. B. Biological Sciences.

[18]  Y. Hosono,et al.  The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model , 1998 .

[19]  Amin Boumenir,et al.  Perron Theorem in the Monotone Iteration Method for Traveling Waves in Delayed Reaction-Diffusion Equations , 2006, math/0612196.

[20]  Shiwang Ma,et al.  Traveling Wavefronts for Delayed Reaction-Diffusion Systems via a Fixed Point Theorem , 2001 .

[21]  Jérôme Coville,et al.  On uniqueness and monotonicity of solutions of non-local reaction diffusion equation , 2006 .

[22]  Xiao-Qiang Zhao,et al.  Dynamical systems in population biology , 2003 .

[23]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[24]  Paul C. Fife,et al.  Propagating fronts for competing species equations with diffusion , 1980 .

[25]  Chunhua Ou,et al.  Persistence of wavefronts in delayed nonlocal reaction-diffusion equations , 2007 .